Short-time dynamics in $s+is$-wave superconductor with incipient bands

Motivated by the recent observation of the time-reversal symmetry broken state in K-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ superconducting alloys, we theoretically study the collective modes and the short time dynamics of the superconducting state with $s+is$-wave order parameter using an effective four-band model with two hole and two electron pockets. The superconducting $s+is$ state emerges for incipient electron bands as a result of hole doping and appears as an intermediate state between $s^{\pm }$ (high number of holes) and $s^{++}$ (low number of holes). The amplitude and phase modes are coupled giving rise to a variety of collective modes. In the $s^{\pm }$ state, we find that the collective excitations are the Higgs (amplitude) modes, while the Leggett mode is absent due to strong interband interaction. In the $s+is$ and $s^{++}$ state, we uncover a new coupled collective soft mode. Finally we compare our results with the $s+id$ solution and find similar behavior of the collective modes as in the $s+is$ state.

Figure 1

Fermi surface sketch for the optimally (a) and overdoped (b) ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ plotted for the folded (black) and unfolded (gray) Brillouin zone consisting of 1 and 2-Fe atoms per unit cell, respectively. Blue circles represent the two hole pockets at the $\mathrm{\Gamma }$ point and the red ellipses represent the two electron pockets at the $M$ Point. In the overdoped regime ($x\sim 1$) the electron pockets shift away from the Fermi level and become incipient.

Figure 2

Solution of the mean-field equations for the pairing amplitudes on the electron and hole pockets shown here as a function of the parameter $E_{Fe}=\mu +E_D/2$ with $E_D=0.12$ eV, $t_e=t_h=2.54$ eV, $U_{\mathrm{hh}}=2.1925$ eV, $U_{\mathrm{eh}}=2.3205$ eV. Note that the width of the $s+is$ region is approximately 1.5 meV.

Figure 3

Results of the numerical solution of the equations of motion (4.5) on the discreet momentum mesh with $N_{k_x}\times N_{k_y}=4096$ points. The values of the coupling parameters are chosen so that the initial state is $s^{\pm }$ near the boundary separating the $s^{\pm }$ and $s+is$ ground states. Depending on the initial value of ${\mathrm{\Delta }}_e$ [panels (a) and (b)] there is a critical time $t^{*}$ on which pairing amplitude on the electron pocket grows signaling the onset of the emergence $s+is$ state. The relative phase oscillation of ${\mathrm{\Delta }}_{h_1}$ and ${\mathrm{\Delta }}_{h_2}$ for ${\mathrm{\Delta }}_e(t=0)=1.0e^{-9}$ is shown in (c).

Figure 4

Numerical solution of the equations of motion with a time dependent vector potential $\mathbf{A}\left(t\right)$. The oscillation of both—the amplitude [panel (a)] and the relative phase [panel (b)] of the order parameters—are dominated by the same frequency ${\omega }_c$. For numerical accuracy we choose a cutoff energy ($\mathrm{\Lambda }=30$ meV) and $U_{hh}=22$ meV and $U_{eh}=20$ meV [60].

Figure 5

The mode ${\omega }_c$ across the full $s+is$ state. The dashed lines mark the borders of the TRSB state. The system parameters are chosen similar to Fig. 4.

Figure 6

Mode frequencies of collective mode at $q=0$ by varying doping parameter $E_{Fe}$: ${\omega }_1$ (blue dashed line) is the overall phase mode, ${\omega }_2$ (red solid line) is the coupled low energy mode. We choose $U_{\mathrm{hh}}=2.1925$ eV and $U_{\mathrm{eh}}=2.3205$ eV.

Figure 7

Big arrows are the gap vectors in a complex plane; small arrows are the mode vectors. Horizontal and vertical axis represent the real and imaginary part. From top to bottom are $s_{\pm }$, $s+is$, and $s_{++}$ state.

Figure 8

Decomposition of $s+is$ mode near the transition between $s^{\pm }$ and $s+is$ (top) and between $s^{++}$ and $s+is$ (bottom).

Figure 9

The solution of the mean-field equations at zero temperature. In this picture the different order parameters are plotted against $E_{Fe}$. Between $-10.5$ meV and $-18.5$ meV both hole order parameters consist out of a s-wave component and a d-wave component. We introduced an energy cutoff $\mathrm{\Lambda }=30$ meV and $U_{hh,s}=20$ meV, $U_{hh,d}=50$ meV and $U_{eh}=22$ meV.

Figure 10

The frequency of the collective mode ${\omega }_c$ over the whole $s+i\mathrm{d}$ regime. Here we also compare the data to $2|{\mathrm{\Delta }}_h^{s,d}|\equiv 2|{\mathrm{\Delta }}_{h_1}^{s,d}|=2|{\mathrm{\Delta }}_{h_2}^{s,d}|$.